The France midfielder swapped the blue of champions Leicester City for Chelsea in a £30million deal last summer, and he could join his compatriot Eric Cantona in winning the league in consecutive seasons for different clubs.
The Star quote a report from French news outlet Le10 Sport, who claim Real have been impressed by Kante's dynamic performances for the Premier League leaders this term.
Kante has started all but one of Chelsea's Premier League games this season, establishing himself as a key player as the club have surged to a seven point lead at the top with just seven games remaining.
"We work with all the players to improve them and to try to work on their strengths and also to work on their weaknesses". So I kept working in my local team Suresnes, a little team in a suburb of Paris. "Tottenham are a good team playing very well, so it's not done and we have to fight".
And the reliable star is not the only Chelsea player on Madrid's radar.
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Given the respective seasons the two have enjoyed in 2016-17, there will likely be huge frustration among Arsenal fans to learn Kante could have been an Arsenal player.
At Boulogne and Caen, the midfielder made up for lost time. "That is why I didn't go to Arsenal and went to Chelsea", Kante said when asked about interest from the Gunners, as quoted by Goal.
'That's why people put it on me but it is not me.
'I am not one of the best midfielders in the world or something like that.
Kante told the Mirror: "When I was young, between 13 to 18, I tried to get into professional academies like Rennes, Lorient, Sochaux maybe four or five different ones".